What are the domain and range of the exponential function below?
[tex]F(x)=4^x+3[/tex]
A. Domain: All real numbers
Range: All real numbers greater than 0
B. Domain: All real numbers
Range: All real numbers greater than 3
C. Domain: All real numbers greater than 0
Range: All real numbers greater than 3
D. Domain: All real numbers greater than 0
Range: All real numbers greater than 0
Answer (1)
The domain of the exponential function F ( x ) = 4 x + 3 is all real numbers.
The range of the exponential function 4 x is all real numbers greater than 0.
Adding 3 to the function shifts the range upwards by 3 units.
The range of F ( x ) = 4 x + 3 is all real numbers greater than 3. Therefore, the answer is B .
Explanation
Understanding the Problem The problem asks us to find the domain and range of the exponential function F ( x ) = 4 x + 3 . The domain is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all possible output values (F(x)-values) that the function can take.
Determining the Domain The domain of an exponential function of the form a x , where a is a positive real number, is all real numbers. This is because we can raise a positive number to any real power. In our case, the base is 4, which is positive, so the domain of 4 x is all real numbers. Adding a constant to the exponential function does not change the domain, so the domain of 4 x + 3 is also all real numbers.
Determining the Range The range of the exponential function 4 x is all real numbers greater than 0. This is because 4 x is always positive for any real number x , and it can take any positive value. When we add 3 to the function, we shift the graph upwards by 3 units. Therefore, the range of 4 x + 3 is all real numbers greater than 3.
Final Answer Therefore, the domain of F ( x ) = 4 x + 3 is all real numbers, and the range is all real numbers greater than 3.
Examples
Exponential functions are used to model various real-world phenomena, such as population growth, radioactive decay, and compound interest. For example, if you invest money in an account that earns compound interest, the amount of money you have after a certain time can be modeled by an exponential function. Understanding the domain and range of these functions helps you determine the possible values of the variables involved, such as the time or the amount of money.
